I am not entirely sure what the question is here...are you basically asking, Let where is some indexing set. Then every subset of can be thought of as a product consisting only of the operations , , and ? I do not think this is true. Let (basically ) and let and . Thus . Then, , and your products will be, . Thus there is no way of getting the elements or to be in a singleton set, which is a contradiction. However, I am unsure if this was actually your question... Also - use LaTeX! \cap is intersection, \cup is union, \{ is left curly bracket while \} is the right. See here. (Also, I believe that and are often used for .)

The partitions which are created by 'n' subsets of S, can be represented as an 'n' digit binary number
say 0100....001
Where ever you have 1 - Take as such, else complement it before taking the intersection.

For the above binary it would be

What you wrote works only in the specific example you gave. (A better way to visualize these partitios is to consider any two subsets of S and see what each of the partition as I define them is in the Venn Diagram)

By exhaustive, I mean, there exits a partition, , (infact a unqiue parition as all the partition are mutually exclusive) such that

The partitions which are created by 'n' subsets of S, can be represented as an 'n' digit binary number
say 0100....001
Where ever you have 1 - Take as such, else complement it before taking the intersection.

For the above binary it would be

What you wrote works only in the specific example you gave. (A better way to visualize these partitios is to consider any two subsets of S and see what each of the partition as I define them is in the Venn Diagram)

Sorry if this is getting more and more confusing - seems I am not able to convey the crux of the problem.

This is what it is
1. Let there be any universal set S
2. Let there be any subsets of this universal set, we call them
3. Based on we define partitions (as in my post above). In general there will be such partitions, some of which may be (null set)
4. Now let X be any set whcih can be constructed from using the operators below
a> Intersection
b> Union
c> Complement
5. Prove that X can be represented as union of partitions (as we defined them in step 3 above)

Also I would write down how I think we can proceed with the proof.
Show that if there are any two subsets of S, A and B where A,B can be expressed as union of the partitions then each of the following can also be expressed as union of paritions
1. A' (or B')
2.
3.

Now to start with each of the subsets can be expressed as sum of partitions hence any X can also be.

What I am struggling with is the case when we make 'n' infinite. Not sure if my logic breaks down somewhere?

Sorry if this is getting more and more confusing - seems I am not able to convey the crux of the problem.

This is what it is
1. Let there be any universal set S
2. Let there be any subsets of this universal set, we call them
3. Based on we define partitions (as in my post above). In general there will be such partitions, some of which may be (null set)
4. Now let X be any set whcih can be constructed from using the operators below
a> Intersection
b> Union
c> Complement
5. Prove that X can be represented as union of partitions (as we defined them in step 3 above)

Also I would write down how I think we can proceed with the proof.
Show that if there are any two subsets of S, A and B where A,B can be expressed as union of the partitions then each of the following can also be expressed as union of paritions
1. A' (or B')
2.
3.

Now to start with each of the subsets can be expressed as sum of partitions hence any X can also be.

What I am struggling with is the case when we make 'n' infinite. Not sure if my logic breaks down somewhere?

Induct on the number of operations you have. That is, you want to show that if you take a union of partitions, A, and union or intersect it with one of your sets, or take its complement, and see if you get a union of partitions. The complement and union are obvious, so concentrate on intersection.

This method clearly does not depend on the number of partitions, but does depend on the number of operations you are performing (so you can't take an infinite intersection or something). If you are going to take an infinite intersection then I'm not actually sure if your result holds (think of open sets in a topological space...).

Yes - it is the infinite case that is troubling me. Anyways thanks for your patience

Which infinite case - the case of infinite products (e.g. taking an infinite intersection) or the case of an infinite number of partitions (which is what your fist post seems to imply). The two cases are different.